You could say both, either 'I just hear the sound of the water dripping' or 'I just hear the sound of the dripping water'. I consider the meaning is the same and both are equally correct and natural. You wouldn't say dropping, if the volume of water were greater than just a drip, 'falling' would be preferred. Hope that helps.

Assuming the first one is a typo and you meant to say "dripping," both are perfectly correct and natural. I think there is a subtle difference in emphasis. It's tiny. It might explain why a speaker might decide to say it one way or the other. "The sound of the water dripping" emphasizes "dripping." It is "the sound of dripping." In your mind, you are thinking "drip... drip... drip..." The fact that it's water is secondary. "The sound of the dripping water" emphasizes water. It is "the sound of water." You think "wet, water, oh dear it's going to damage the house." The fact that the water is dripping is secondary.

I can only hear the sound of the dripping water. You wouldn't use "dropping" for water in this case unless it's individual drops of water.

I just hear the sound of dripping water sounds more natural